Some problems in harmonic analysis on quantum …swang/beamer...Sidon sets and ( p)-sets 2 of 41...

Universite de Franche-Comte - Polish Academy of Sciences

Some problems in harmonic analysison quantum groups

Supervisors

Quanhua Xu

Adam Skalski

Candidate

Simeng Wang

June 22, 2016

Outline

Introduction to abstract harmonic analysis on quantum groups

Convolutions of states and Lp-improving operators

Sidon sets and Λ(p)-sets

2 of 41

Simeng Wang - Some problems in harmonic analysis on quantum groups

Outline




3 of 41


Classical frameworks

Abstract harmonic analysis studies Fourier series, integrals and relatedproblems on topological groups. There are two important frameworksstudied in recent decades:• Harmonic analysis on compact groups• Harmonic analysis on group algebras of discrete groups

Observation:• G a compact group comultiplication on C (G ):

∆G : C (G )→ C (G × G ), ∆G (f )(s, t) = f (st), s, t ∈ G .(C (G ),∆G ) determines G .

• Γ a discrete group group C*-algebra C ∗r (Γ) with comultiplication∆C∗r (Γ) : C ∗r (Γ)→ C ∗r (Γ)⊗ C ∗r (Γ). (C ∗r (Γ),∆C∗r (Γ)) determines Γ.

(If Γ is abelian, we have a compact Pontryagin dual group Γ with

(C (Γ),∆Γ) ∼= (C∗r (Γ),∆C∗

r (Γ)).)

4 of 41


Classical frameworks

Abstract harmonic analysis studies Fourier series, integrals and relatedproblems on topological groups. There are two important frameworksstudied in recent decades:• Harmonic analysis on compact groups• Harmonic analysis on group algebras of discrete groups

Observation:• G a compact group comultiplication on C (G ):

∆G : C (G )→ C (G × G ), ∆G (f )(s, t) = f (st), s, t ∈ G .(C (G ),∆G ) determines G .

• Γ a discrete group group C*-algebra C ∗r (Γ) with comultiplication∆C∗r (Γ) : C ∗r (Γ)→ C ∗r (Γ)⊗ C ∗r (Γ). (C ∗r (Γ),∆C∗r (Γ)) determines Γ.

(If Γ is abelian, we have a compact Pontryagin dual group Γ with

(C (Γ),∆Γ) ∼= (C∗r (Γ),∆C∗

r (Γ)).)4 of 41


Woronowicz’s Compact quantum groups

We may extend the previous arguments in a more general framework,that is, compact quantum groups.

• A compact quantum group is a pair G = (A,∆), where:

A : a unital C*-algebra, ∆ : A→ A⊗ A a *-homomorphism s.t.

(∆⊗ id)∆ = (id⊗∆)∆,

span((1⊗ A)∆(A)) = span((A⊗ 1)∆(A)) = A⊗ A.

A := C (G) is called the algebra of “continuous functions” on G.

• There exists a Haar state h on C (G) which is “translate invariant”.

5 of 41


Towards Fourier analysis: the “dual group”

Recall the Fourier transform on a compact abelian group G ,

F : functions on G → functions on Pontryagin dual G .

- G non-abelian: replace G by Irr(G ) (irreducible representations)

- For a compact quantum group G: Irr(G) ?

• Unitary representation of G: u = [uij ]ni ,j=1 ∈Mn(C (G)) unitary s.t.

∀1 ≤ j , k ≤ n, ∆(ujk) =∑n

p=1ujp ⊗ upk .

Irr(G): equivalent classes of all such irreducible representations u.For each π ∈ Irr(G) choose a representative u(π).

• All such matrix coefficients u(π)ij (π ∈ Irr(G)) span a dense algebra

of “polynomials” Pol(G) ⊂ C (G).

6 of 41



Recall the Fourier transform on a compact abelian group G ,

F : functions on G → functions on Pontryagin dual G .

- G non-abelian: replace G by Irr(G ) (irreducible representations)- For a compact quantum group G: Irr(G) ?• Unitary representation of G: u = [uij ]

ni ,j=1 ∈Mn(C (G)) unitary s.t.

∀1 ≤ j , k ≤ n, ∆(ujk) =∑n

p=1ujp ⊗ upk .

Irr(G): equivalent classes of all such irreducible representations u.For each π ∈ Irr(G) choose a representative u(π).

• All such matrix coefficients u(π)ij (π ∈ Irr(G)) span a dense algebra

of “polynomials” Pol(G) ⊂ C (G).

6 of 41



In the framework of locally compact quantum groups, there is a“dual” discrete quantum group, denoted G, subject to the following∗-algebra

cc(G) := ⊕π∈Irr(G)Mnπ(C).

We may define the Fourier transform

F : Pol(G)→ cc(G), x 7→ x ,

where

x(π) = (h ⊗ id)((u(π))∗(x ⊗ 1)) = [h(u(π)ji

∗x)]nπi ,j=1, π ∈ Irr(G).

( Recall: for a compact group G , f ∈ C (G ),

f (π) =

∫Gu(π)(g)∗f (g)dm(g) =

[∫Gu

(π)ji

∗fdm

]nπi ,j=1

.)

7 of 41


Lp-Fourier series

Briefly, we obtain the Fourier transform

F : Pol(G)→ cc(G) := ⊕π∈Irr(G)Mnπ(C), x 7→ x .

The map can be extended to Lp-spaces.

• (πh,Hh) = GNS of Pol(G) wrt h, L∞(G) = πh(Pol(G))′′ ⊂ B(Hh).

• Define ‖x‖1 = ‖h(·x)‖L∞(G)∗ for x ∈ Pol(G), and let L1(G) be thecompletion of (Pol(G), ‖‖1). Define Lp(G) to be the complexinterpolation space

Lp(G) = (L∞(G), L1(G))1/p, 1 ≤ p ≤ ∞.

• Define `p(G) on cc(G) similarly.

8 of 41


Lp-Fourier series

Briefly, we obtain the Fourier transform

F : Pol(G)→ cc(G) := ⊕π∈Irr(G)Mnπ(C), x 7→ x .

The map can be extended to Lp-spaces.

• (Hausdorff-Young inequality) We have the extension

F : Lp(G)→ `q(G) contraction, 1/p + 1/q = 1.

• (Plancherel theorem) F : L2(G)→ `2(G) unitary.

9 of 41


More remark on G: non-unimodularity

Recall: a discrete group is always unimodular.But the discrete quantum group G can be non-unimodular.

There are different “left/right invariant” Haar weights on `∞(G), anda modular element F linking them,

F = (Fπ)π∈Irr(G), Fπ ∈Mnπ(C).

It is possible ‖F‖ = supπ ‖Fπ‖ = +∞.

In fact, F is trivial iff h on L∞(G) is tracial. Woronowicz used this Fto implement the modular property of non-tracial Haar state h on G.

10 of 41


Convolutions

• Recall: For a compact group G and µ1, µ2 ∈ M(G ),

∀f ∈ C (G ),

∫Gfd(µ1 ? µ2) =

∫G×G

f (gh)dµ1(g)dµ2(h).

If dµ1 = f1dm, dµ2 = f2dm with f1, f2 ∈ L1(G ), then “define”

(f1 ? f2)dm = (f1dm) ? (f2dm).

• Similarly, for a CQG G and ϕ1, ϕ2 ∈ L∞(G)∗, we may define

ϕ1 ? ϕ2 = (ϕ1 ⊗ ϕ2) ◦∆.

This also induces the convolution x1 ? x2 for x1, x2 ∈ L1(G):

h(·x1 ? x2) = h(·x1) ? h(·x2).

11 of 41


Convolutions on Lp-spaces

The modular element F gives rise to a scaling group τ of∗-automorphisms of L∞(G) such that

τt(u(π)) = F it

π u(π)F it

π , π ∈ Irr(G).

Young’s inequality (Liu-W.-Wu, 2015): for x , y ∈ Pol(G) we have

‖τ ip′

(y) ? x‖r ≤ ‖x‖p‖y‖q,1

r+ 1 =

1

p+

1

q,

1

p+

1

p′= 1

where 1 ≤ p, q, r ≤ ∞ if h is tracial, and 1 ≤ p, q, r ≤ 2 for generalcases (counterexample existing for r =∞).Remark: similar for LCQGs

12 of 41


Fourier multipliers

• Multipliers: Each a = (aπ)π∈Irr(G) ∈∏π∈Irr(G) Mnπ induces a

mapma : Pol(G)→ Pol(G), max = F−1(ax).

We say a is a left bounded multiplier on Lp(G) if ma extends to abdd map on Lp(G). (similar def. for right multipliers)M(Lp(G)) ={left & right bdd multipliers on Lp(G)} ⊂ B(Lp(G)).

• Daws, Neufang, Junge, Ruan (09’-12’): study completely boundedmultiplier on L∞(G). Easy to establish

‖a‖`∞(G) ≤ ‖a‖Mcb(L∞(G)).

13 of 41


Fourier multipliers

• Multipliers: Each a = (aπ)π∈Irr(G) ∈∏π∈Irr(G) Mnπ induces a

mapma : Pol(G)→ Pol(G), max = F−1(ax).

We say a is a left bounded multiplier on Lp(G) if ma extends to abdd map on Lp(G). (similar def. for right multipliers)M(Lp(G)) ={left & right bdd multipliers on Lp(G)} ⊂ B(Lp(G)).

Proposition (Partially communicated by M. Junge)

‖F14− 1

2p aF−14

+ 12p ‖`∞(G) ≤ ‖a‖M(Lp(G)).

14 of 41


Outline




15 of 41


Background

A classical question: contruction of positive Borel probablitymeasures µ so that for a given p > 1,

∃p < q, ∀f ∈ Lp(T,dm

2π), ‖µ ? f ‖q ≤ ‖f ‖p.

Oberlin (1982): the Cantor-Lebesgue measure supported by theusual middle-third Cantor set satisfies the above property.In fact it suffices to show

∃p < 2, ‖µ ? f ‖2 ≤ ‖f ‖p, f ∈ Lp(Z/3Z,P)

where P is the normalized counting measure, µ({0}) = µ({2}) = 12 .

16 of 41


Background

Motivated by Oberlin, Ritter (1984) showed that:

Theorem

If G is an arbitrary finite group and Tµ : f 7→ µ ? f is the convolutionoperator associated to a probability measure µ on G , then(∃ p < 2, ‖Tµ : Lp(G )→ L2(G )‖ = 1

)⇔ G = 〈ij−1 : i , j ∈ suppµ〉.

The operator T with ‖T : Lp → L2‖ = 1 for a p < 2 will be said tobe Lp-improving throughout the presentation.

Motivating question

Similar constructions on (finite/compact) quantum groups?

17 of 41


Interpretation in the quantum language

Recall the classical result: for Tµ : f 7→ µ ? f , f ∈ C (G )(∃ p < 2, ‖Tµ : Lp(G )→ L2(G )‖ = 1

)⇔ (∗) G = 〈ij−1 : i , j ∈ suppµ〉.

Take ν = µ(·−1) ? µ. It is a simple observation that

(∗)⇔ ∀f > 0, ∃n ≥ 1,

∫fdν?n > 0.

This corresponds to an important notion in the study of random walkson quantum groups.

Definition (Vaes,Vergnioux 07; Kalantar,Neufang,Ruan 14)

A state ψ on C (G) is non-degenerate if

∀0 6= x ∈ C (G)+, ∃n ≥ 1, ψ?n(x) > 0.18 of 41


Main result

Theorem (W.)

Let G be a finite quantum group and ϕ be a state on C (G). Denoteψ = (ϕ ◦ S) ? ϕ. Then

∃1 < p < 2, ∀ x ∈ C (G), ‖ϕ ? x‖2 ≤ ‖x‖p

iff ψ is non-degenerate: ∀0 6= x ∈ C (G)+,∃n ≥ 1, ψ?n(x) > 0.

19 of 41


Infinite case: a free product construction

Let G1 and G2 be two CQGs. There exist a CQG G := G1∗G2 s.t.

C (G) = C (G1) ∗ C (G2), L∞(G) = L∞(G1)∗L∞(G2).

Theorem (W.)

Let G1, . . . ,Gn be finite quantum groups and let each ϕi be a stateon C (Gi ), i ∈ {1, . . . , n}. Take G = G1∗ · · · ∗Gn and denote byϕ = ϕ1 ∗ · · · ∗ ϕn the conditionally free product. If for each i ,

∃1 ≤ p < 2 s.t. ∀ x ∈ C (G), ‖ϕi ? x‖2 ≤ ‖x‖p,

then the convolution operator T : x 7→ ϕ ? x , x ∈ C (G) satisfies

∃1 < p′ < 2, ‖T : Lp′(G)→ L2(G)‖ = 1.

20 of 41


Key ingredient I: Lp-improvement vs spectral gap

Theorem (W.)

Let A be a finite dimensional C*-algebra equipped with a faithfultracial state τ , and T : A→ A be a unital 2−positive trace preservingmap on A. Then

∃1 < p < 2, ‖T : Lp(A)→ L2(A)‖ = 1

if and only if

supx∈A\{0},τ(x)=0

‖Tx‖2

‖x‖2

(= sup‖x‖2=1,τ(x)=0

〈|T |x , x〉1/2L2(A,τ)

)< 1.

Also, the Lp-improving property is stable under free product.21 of 41


Key ingredient II: non-degenerate states, applications

Proposition (W.)

If ϕ is a non-degenerate state on C (G), then

w∗- limn→∞

1

n

n∑k=1

ϕ?k = h.

This proposition yields useful results for studying quantum subgroups(→ next slides).

22 of 41



Recently, Skalski-So ltan discussed the quantum subgroups “generatedby a quantum subset” via the Hopf image of Banica-Bichon. Each∗-representation π of C (G) associates a quantum subgroup Gπ ⊂ G.If G is a group and X ⊂ G a closed subset with the restriction mapπ : C (G )→ C (X ), then Gπ = 〈X 〉.

Corollary (Banica,Franz,Skalski 12; W.)

Let A be a unital C*-algebra with a unital ∗-homomorphismπ : C (G)→ A, and q : C (G)→ C (Gπ) the quotient map. Thengiven any faithful state ϕ on A,

hGπ ◦ q = w∗- limn→∞

1

n

n∑k=1

(ϕ ◦ π)?k .

23 of 41



This result is recently applied in a series of works of Banica on matrixmodels of quantum groups. For instance,

• (Banica 16) If G ⊂ O∗n , then we have a matrix modelC (G) ⊂M2(C (H)) (H a compact Lie group) such that the Haarstate of G coincides with tr ⊗

∫.

• (Banica 16) We have a Weyl matrix modelπ : C (S+

N )→MN(C (H)) (H ⊂ Un) such that the image of π is aHopf algebra and the associated Haar state coincides with tr ⊗

∫.

• (Banica 16; Banica-Nechita 16) For the deformed Fourier matrixmodel π : C (S+

MN)→MMN(C (TMN)), the law of the maincharacter of the Hopf image is asymptotically free Poisson.

24 of 41


Outline




25 of 41


Sidon sets: classical settings

Definition

Let G be a compact abelian group and Γ = G be the dual discretegroup. A subset E ⊂ Γ is called a Sidon set if

∀(αγ) ⊂ C,∑γ⊂E|αγ | ∼ ‖

∑γ∈E

αγγ‖L∞(G).

• Various characterizations: interpolation of bounded measures,multipliers, Λ(p)-estimations, unconditional bases...

• Typical examples: Rademacher functions; lacunarity in Z...

26 of 41



Definition

Let G be a compact abelian group and Γ = G be the dual group. Asubset E ⊂ Γ is called a Sidon set if

∀(αγ) ⊂ C,∑γ⊂E|αγ | ∼ ‖

∑γ∈E

αγγ‖L∞(G).

Noncommutative generalizations:

• G compact non-abelian group, Γ Irr(G );

• Γ discrete non-abelian group, L∞(G ) L∞(Γ) = VN(Γ) groupvNa. (Γ the dual quantum group of Γ, Irr(Γ) = Γ)

27 of 41



Let G be a compact group.For f ∈ L∞(G ) and π ∈ Irr(G ), recall f (π) =

∫G f (g)u(π)(g)∗dm(g),

‖f ‖1 =∑

π∈Irr(G)

dπTr(|f (π)|).

Theorem (Figa-Talamanca)

Consider E ⊂ Irr(G ). TFAE:

1. E is a Sidon set, i.e., supp(f ) ⊂ E ⇒ ‖f ‖1 ∼ ‖f ‖∞;

2. ⊕π∈EMnπ = {µ|E : µ ∈ M(G )};3. ⊕c0

π∈EMnπ = {f |E : f ∈ L1(G )}.

28 of 41


Sidon sets: quantum group setting

Let G be a compact quantum group. F the modular element for G.For x ∈ L∞(G), π ∈ Irr(G), recall x(π) = (h ⊗ id)((u(π))∗(x ⊗ 1)),

‖x‖1 =∑

π∈Irr(G)

dπTr(|x(π)Fπ|). (dπ = Tr(Fπ))

Theorem (W.)

Consider E ⊂ Irr(G). TFAE:

1. E is a Sidon set, i.e., supp(x) ⊂ E ⇒ ‖x‖1 ∼ ‖x‖∞,

2. ⊕π∈EMnπ = {ϕ|E : ϕ ∈ Cr (G)∗};3. ⊕c0

π∈EMnπ = {x |E : x ∈ L1(G)}.

29 of 41


Sidon sets: classical settings revisited

The previous theorem improves a result of Picardello: (Picardelloproved it in the special case that Γ is amenable)

Corollary

Let Γ be a discrete group (not necessarily amenable). TFAE:

1. E ⊂ Γ is a Sidon set, i.e.,

∀(αγ) ⊂ C,∑γ∈E|αγ | ∼ ‖

∑γ∈E

αγλ(γ)‖VN(Γ);

2. E ⊂ Γ is a strong Sidon set, i.e.,

c0(E ) = {f |E : f ∈ A(Γ)(∼= L1(Γ))}.

30 of 41


Remark: various generalizations vs amenability

• G is called coamenable if ε : u(π)ij 7→ δij is bdd on L∞(G).

Recall our convention: Γ = (C ∗r (Γ),∆C∗r (Γ)) being a compact

quantum group. Γ is coamenable iff Γ is amenable.

• If G is not coamenable, there are various (non-equivalent)analogue of Sidon sets: weak Sidon sets, interpolation sets ofmultipliers, unconditional Sidon sets, Leinert sets, etc.(For Γ: Pisier 95’)

31 of 41


Sidon sets ⇒ Λ(p)-sets

Definition

E ⊂ Irr(G) is a Λ(p)-set if

‖x‖p ∼ ‖x‖1, supp(x) ⊂ E .

It is well-known that if G is a compact group or the compact dual of adiscrete group, then any Sidon set for G is a Λ(p)-set for 1 < p <∞.• Case for compact groups: Figa-Talamanca, Marcus-Pisier;• Case for (dual of) discrete groups: Picardello, Harcharras...• Case for compact quantum groups: more delicate –

Blendek-Michalicek 13’: Helgason-Sidon ⇒ central Λ(2) in Kaccase.

32 of 41


Sidon sets ⇒ Λ(p)-sets, Fourier multipliers

Theorem (W.)

Let 2 < p <∞ and E ⊂ Irr(G). Then:

‖x‖p ∼ ‖x‖1, supp(x) ⊂ E ,

if and only if

∀ a ∈ ⊕π∈EMnπ , ∃ b ∈ M(Lp(G)), b|E = a.

Remark: the completely bdd version of above thm is not establishedyet.(For G = Γ dual of discrete group: Harcharras 99’;For G = G compact group: Hare-Mohanty 15’.)

33 of 41


Sidon sets ⇒ Λ(p)-sets, Fourier multipliers

Theorem (W.)

Let 2 < p <∞ and E ⊂ Irr(G). Then:

‖x‖p ∼ ‖x‖1, supp(x) ⊂ E ,

if and only if

∀ a ∈ ⊕π∈EMnπ , ∃ b ∈ M(Lp(G)), b|E = a.

Corollary

Any Sidon set for G is a Λ(p)-set for 1 < p <∞.

34 of 41


Observations on non-unimodularity I

Recall that we have a modular element F for G.

Theorem

Let E ⊂ Irr(G) a Λ(p)-set for 2 < p <∞. Then

supπ∈E‖Fπ‖ < +∞.

Drinfeld-Jimbo deformation: a compact semi-simple Lie group G acompact quantum group Gq (0 < q < 1), with Gq non-unimodular.

Corollary

SU(2)q does not admit infinite Λ(p)-set for any 2 < p <∞.

35 of 41


Observations on non-unimodularity II

More words on noncommutative Lp:Let h be non-tracial on G (equivalently G non-unimodular)

• According to Kosaki 84’: For each 0 ≤ θ ≤ 1, there is a complexinterpolation scale (Lp(θ)(G))1≤p≤∞ between L∞(G) and L∞(G)∗,which are isometrically isomorphic but not equal.

In the previous slides we have indeed taken Lp(G) = Lp(0)(G).

• Casper 13’: The boundedness of Fourier transform depends on θ.

36 of 41


Observations on non-modularity II

Proposition

Our definition of Λ(p)-sets is independent of θ.That is, let 2 < p <∞, 0 ≤ θ, θ′ ≤ 1 and E ⊂ Irr(G). Then:

‖x‖Lp(θ)

(G) ∼ ‖x‖L1(θ)

(G), supp(x) ⊂ E ,

if and only if

‖x‖Lp(θ′)(G) ∼ ‖x‖L1

(θ′)(G), supp(x) ⊂ E .

37 of 41


Existence of Λ(p)-sets

Theorem (Bozejko; W.)

Let (M, ϕ) be a vNa equipped with a normal faithful state ϕ. LetB = {xi ∈ M : i ≥ 1} be an orthogonal system wrt ϕ s.t.supi ‖xi‖∞ <∞. Then for each 1 ≤ p <∞, there exists an infinitesubset {xik : k ≥ 1} ⊂ B

∀(ck) ⊂ C, ‖∑k≥1

ckxik‖Lp(M) ∼(∑

k≥1

|ck |2) 1

2.

38 of 41


Existence of Λ(p)-sets

Corollary

Let G be a CQG. Let E ⊂ Irr(G) be an infinite subset withsupπ∈E dπ <∞. Then for each 1 ≤ p <∞, there exists an infinitesubset E ′ ⊂ E s.t.

‖x‖p ∼ ‖x‖1, supp(x) ⊂ E ′.

39 of 41


Some examples

•∏

k Gk with ONk⊂ Gk ⊂ U+

Nk: the collection of fundamental

representations of all Gk gives a (weak) Sidon set and a Λ(p)-setfor 1 < p <∞.

•∏

k SUqk (2): the collection of fundamental representations of allSUqk (2) gives a Sidon set and a Λ(p)-set for 1 < p <∞ iffinfk qk > 0.

• O+N admits no infinite central Λ(p)-set for p > 4; but we can

construct a concrete central Λ(4)-set for SUq(2).

• For any simply connected compact semi-simple Lie group G , theDrinfeld-Jimbo deformation Gq does not admit infinite (central)Sidon set.

40 of 41


Thank you very much!

41 of 41


Some problems in harmonic analysis on quantum …swang/beamer...Sidon sets and ( p)-sets 2 of 41...

Documents

Transcript of Some problems in harmonic analysis on quantum …swang/beamer...Sidon sets and ( p)-sets 2 of 41...